There is a flashback episode with President Bridges, and VP Allen flying on Air Force One. The President and VP are not allowed to fly on the same air plane.
In episode#1.7, "Saving the Century", the Speaker of the House is overseeing confirmation hearings for the Vice President. The SOTH does not sit on any committees, nor does he actively vote in House business. In the past the House Judicial Committee has held hearings for VP Confirmation.